Problem 275Published on Friday, 22nd January 2010, 05:00 pm; Solved by 403; Difficulty rating: 85%
Let us define a balanced sculpture of order n as follows:
- A polyomino made up of n+1 tiles known as the blocks (n tiles)
and the plinth (remaining tile);
- the plinth has its centre at position (x = 0, y = 0);
- the blocks have y-coordinates greater than zero (so the plinth is the unique lowest tile);
- the centre of mass of all the blocks, combined, has x-coordinate equal to zero.
When counting the sculptures, any arrangements which are simply reflections about the y-axis, are not counted as distinct. For example, the 18 balanced sculptures of order 6 are shown below; note that each pair of mirror images (about the y-axis) is counted as one sculpture:
There are 964 balanced sculptures of order 10 and 360505 of order 15.
How many balanced sculptures are there of order 18?